Xref: utzoo comp.ai:6163 sci.philosophy.tech:2187 Path: utzoo!attcan!uunet!jarthur!usc!venera.isi.edu!smoliar From: smoliar@vaxa.isi.edu (Stephen Smoliar) Newsgroups: comp.ai,sci.philosophy.tech Subject: Re: Another letter to the New York Review Message-ID: <12240@venera.isi.edu> Date: 6 Mar 90 15:34:35 GMT Sender: news@venera.isi.edu Reply-To: smoliar@vaxa.isi.edu.UUCP (Stephen Smoliar) Organization: USC-Information Sciences Institute Lines: 68 Summary: In article <90Mar3.152728est.6160@neat.cs.toronto.edu> radford@ai.toronto.edu (Radford Neal) writes: > >Let's say that one day Penrose announces that he is able to solve, say, >the word problem for semi-groups - a well-known non-computable problem. >People give him instances of this problem. After a period of time >that goes up only reasonably with the size of the instance he announces >the answer: YES or NO. In those cases where the true answer is >subsequently determined, he always turns out to be right. This holds >even for very difficult cases that require increasingly subtle >arguments to establish that the answer is NO, as well as cases where >extremely complex reductions are needed to demonstrate that the answer >is YES. > I think this gets to the heart of the matter behind Penrose's argument very nicely. It also raises some rather troubling issues which probably need further discussion. I realize that, to some extent, these issues conflict with a personal tendency to try to abstract the world into deductive operations; but, in spite of the fact that I know this may only be a personal bias, there are aspects of this example which worry me. I think the problem is one of the epistemological foundations of issues of BELIEF. Penrose announces that he can solve the word problem. For any case we give him, he responds with the correct answer. Has he convinced us? Let me illustrate the difficulty by constructing what I hope is a valid analogy: Penrose announces that he can solve the general medical diagnosis problem. For any case from the medical literature we give him, he responds with the correct answer. If you were running a hospital, would you want him going on rounds through your wards? The point I am trying to get at here is that there is more to reasoning than getting the right answer. If Penrose says (using whatever reasoning powers "work" for him), "This patient has a malignant brain tumor which will be fatal if not corrected by surgery within one hour," would YOU rush the patient into the operating room WITHOUT ASKING ANY QUESTIONS? Whether you are an administrator or a fellow physician, chances are, you would want some kind of JUSTIFICATION for Penrose's decision before taking any hasty action; and if, when confronted with the question of justification, Penrose were to respond, "I just know it," you would be in quite a quandary. (Believe me, I have no idea how I would respond in such a situation, particularly if I knew that Penrose's track record had been flawless prior to this incident.) I think the point we have to confront here is that CONVINCING reasoning lies at neither extreme. If we are talking about "the process by which real mathematicians [or any other thinkers, for that matter] establish new truths," then we have no reason to believe that deductive systems are the only mechanisms which may kick in. Instead, as Marvin Minsky pointed out in THE SOCIETY OF MIND, such systems for formal reasoning are better qualified to SUMMARIZE and JUSTIFY those "new truths," once they have been encountered. The question then reduces to whether or not there are MECHANISMS, deductive or otherwise, for establishing (or even hypothesizing) them. AI argues that such mechanisms do, indeed, exist; but, because we have given so much of our attention to deductive systems, we are probably still a far cry from a better understanding of their nature. ========================================================================= USPS: Stephen Smoliar USC Information Sciences Institute 4676 Admiralty Way Suite 1001 Marina del Rey, California 90292-6695 Internet: smoliar@vaxa.isi.edu "Only a schoolteacher innocent of how literature is made could have written such a line."--Gore Vidal